The systematic study of stochastic optimization problems under changes to probability distributions extends at least back to [1]}, [2]}, [3]}, [4]}; see [5]} and [6]} for more recent efforts in the context of convex problems and stochastic dominance, respectively. There is an extensive literature on how to address parameter uncertainty in a conservative manner; see [7]}, [8]}, [9]} for results from semi-infinite programming and [10]}, [11]}, [12]} for robust convex optimization. Under distributional ambiguity, the focus tends to be on achieving a solution that is “good” across a set of candidate distributions, for example centered at a presently available empirical distribution [13]}, [14]}, [15]}, [16]}, [17]}, [18]}. Related risk-based approaches have similar effect, at least for monotone and positively homogeneous regular measures of risk [19]}, [20]}, [21]}. Adversarial statistical learning addresses ambiguity about training data through conservative perturbations of the support of the corresponding (empirical) distribution [22]}.
Robust regularization [23]} achieves conservativeness through perturbation of a decision vector, with applications to manufacturing [24]} and statistical learning [25]}, [26]}, [27]} in the form of diametrical risk minimization. When approximations stem from empirical distributions, these approaches are well-motivated by the downward bias in sample average approximations [27]}.
We step away from conservative approaches, which correspond to being “pessimistic” about values of ambiguous parameters and distributions, and adopt an “optimistic” approach based on problem relaxation.
